laser - How does optical phase modulation produce photons with different frequencies?

The classical description of electro-optic modulators is an index of refraction that depends on the applied voltage. For example, for a sine modulation $\sin(\Omega t)$, a monochromatic laser of frequency $\omega$ would get an additionnal phase $\varphi\propto\sin(\Omega t)$. This results in sidebands in the spectrum at $\omega-\Omega$ and $\omega+\Omega$.

Now, what is the interpretation of this phenomenon in terms of photons? A photon with initial frequency $\omega$ will end up at $\omega-\Omega$ or $\omega+\Omega$. How can the time-variation of the refractive index create new photon frequencies? Is it a non-linear effect similar to second-harmonic generation? If yes, it could be explained by an interaction such as $\hbar\omega + \hbar\omega \rightarrow \hbar (\omega-\Omega ) + \hbar(\omega+\Omega)$?

EDIT: A corollary to the original question. I shake my hand very fast in front of a laser beam, what happens to the photons? Do they get chopped in shorter photons? Instead of my hands, I could use a super-fast chopper. I would see photons with new frequencies (the sidebands) because of this modulation. How come the incident photons get a different energy?